Homoclinic based on orthogonal collocation

We compute Ntst time slices of a periodic orbit using orthogonal collocation. This is implemented in the structure BifurcationKit.PeriodicOrbitOCollProblem.

The general method is explained in BifurcationKit.jl.

General method

Please see ^[DeWitte] for a thorough description of the method. It amounts to solving a boundary value problem.

\[\left\{\begin{aligned} & \dot{u}(t)-2 T\cdot F(u(t), p)=0 \\ & F\left(u_0, p\right)=0 \\ & Q^{U^{\perp}, \mathrm{T}}\left(u(0)-u_0\right)=0, \\ & Q^{S^{\perp}, \mathrm{T}}\left(u(1)-u_0\right)=0 \\ & T_{22 U} Y_U-Y_U T_{11 U}+T_{21 U}-Y_U T_{12 U} Y_U=0, \\ & T_{22 S} Y_S-Y_S T_{11 S}+T_{21 S}-Y_S T_{12 S} Y_S=0 \\ & \left\|u(0)-u_0\right\|-\epsilon_0=0 \\ & \left\|u(1)-u_0\right\|-\epsilon_1=0 \\ & \int_0^1 \tilde{u}^*(t)[u(t)-\tilde{u}(t)] d t=0, \\ \end{aligned}\right.\]

Mesh adaptation

The goal of this functionality is to adapt the mesh in order to minimise the error.

Jacobian

The jacobian is computed with automatic differentiation e.g. ForwardDiff.jl

References